こんにちは。今回は定期テストはもちろん, それ以外でも頻出の問題をやってみましょう。実際に問題を解いてみてください。解法はそれから見てください。
【問題】四面体OABCにおいて, 辺ABを2 : 1に内分する点をD, 線分CDを3 : 2に内分する点をP, 辺OAの中点をMとする。また, OPと△MBCとの交点をQとする。,
,
とするとき, 次の問いに答よ。
(1) を
,
,
を用いて表せ。
(2) を
,
,
を用いて表せ。
(3) を求めよ。
【解答】
(1)
![Rendered by QuickLaTeX.com \overrightarrow{\mathstrut \text{OD}}=\dfrac13\overrightarrow{ \mathstrut a}+\dfrac23\overrightarrow{ \mathstrut b}\cdots\maru1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-72b44daaa4dcf19ed320b4875c95b5be_l3.png)
![Rendered by QuickLaTeX.com \overrightarrow{ \mathstrut \text{OP}}=\dfrac35\overrightarrow{ \mathstrut \text{OD}}+\dfrac25\overrightarrow{ \mathstrut c}\cdots\maru2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-e9f8718664ca8d221f5b622d789eea63_l3.png)
![Rendered by QuickLaTeX.com \maru2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b55f3d993b1bbf2d3a2ca1e85ea19bd7_l3.png)
![Rendered by QuickLaTeX.com \maru1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c8b8ec9c0d15342374d474f3407d687d_l3.png)
![Rendered by QuickLaTeX.com \overrightarrow{ \mathstrut \text{OP}}=\dfrac35\left(\dfrac13\overrightarrow{ \mathstrut a}+\dfrac23\overrightarrow{ \mathstrut b}\right)+\dfrac25\overrightarrow{ \mathstrut c}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-10e4b6ac0b1420d6841ebbe49257a533_l3.png)
よって,
![Rendered by QuickLaTeX.com \overrightarrow{ \mathstrut \text{OP}}=\dfrac15\overrightarrow{ \mathstrut a}+\dfrac25\overrightarrow{ \mathstrut b}+\dfrac25\overrightarrow{ \mathstrut c}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-8dc906c75612b321d12d040d127a8f53_l3.png)
(2) O, Q, Pは一直線上にあるので,
![Rendered by QuickLaTeX.com \overrightarrow{ \mathstrut \text{OQ}}=k\overrightarrow{ \mathstrut \text{OP}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c8cefcae0060db1303cc8eb37d207a2e_l3.png)
![Rendered by QuickLaTeX.com k](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3e55731b2ccdac9bcef26d20d0e6fd79_l3.png)
とおける。
したがって,
![Rendered by QuickLaTeX.com \overrightarrow{ \mathstrut \text{OQ}}= \dfrac15k\overrightarrow{ \mathstrut a}+\dfrac25k\overrightarrow{ \mathstrut b}+\dfrac25k\overrightarrow{ \mathstrut c}\cdots\maru1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-e50a1b13a2c1c9a9402e0065d50ca3aa_l3.png)
となる。
4点M, B, C, Qは同一平面上にあるから,
![Rendered by QuickLaTeX.com \overrightarrow{ \mathstrut \text{MQ}}=s\overrightarrow{ \mathstrut \text{MB}}+t\overrightarrow{ \mathstrut \text{MC}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-4d1f022973e28889914439be13d6c4fc_l3.png)
![Rendered by QuickLaTeX.com s, t](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-e368057ea2f8371100ea6ca13a6af19d_l3.png)
![Rendered by QuickLaTeX.com \cdots\maru2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-dae7060ff98c074ff0862930c9b3cad6_l3.png)
と表せる。
ここで,
![Rendered by QuickLaTeX.com \begin{array}{lll}\overrightarrow{ \mathstrut \text{MB}}&=&-\overrightarrow{ \mathstrut \text{OM}}+\overrightarrow{ \mathstrut \text{OB}}\\&=&-\dfrac12\overrightarrow{ \mathstrut a}+\overrightarrow{ \mathstrut b}\cdots\maru3\end{array}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-6fac9c2d0a03637a2d6c566079fe1386_l3.png)
また,
![Rendered by QuickLaTeX.com \begin{array}{lll}\overrightarrow{ \mathstrut \text{MC}}&=&- \overrightarrow{ \mathstrut \text{OM}}+\overrightarrow{ \mathstrut \text{OC}}\\&=&-\dfrac12\overrightarrow{ \mathstrut a}+\overrightarrow{ \mathstrut c}\cdots\maru4\end{array}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-4f59e0b2dc6f5f8c732b2ce8d341c418_l3.png)
![Rendered by QuickLaTeX.com \maru2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b55f3d993b1bbf2d3a2ca1e85ea19bd7_l3.png)
![Rendered by QuickLaTeX.com \maru3](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-59e0023e63fc0665bd608c9e370731c9_l3.png)
![Rendered by QuickLaTeX.com \maru4](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-759f766af6f2a34c5000079a99ce3363_l3.png)
![Rendered by QuickLaTeX.com \overrightarrow{ \mathstrut \text{MQ}}=\dfrac12(-s-t)\overrightarrow{ \mathstrut a}+s\overrightarrow{ \mathstrut b}+t\overrightarrow{ \mathstrut c}\cdots\maru5](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3770c4cd5ea5e372d76d78f18fcf7e49_l3.png)
ここで,
![Rendered by QuickLaTeX.com \overrightarrow{ \mathstrut \text{OQ}}=\overrightarrow{ \mathstrut \text{OM}}+ \overrightarrow{ \mathstrut \text{MQ}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-20f2d3e516b18419ca46bd723b048968_l3.png)
![Rendered by QuickLaTeX.com \maru5](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-fb8562820b75568b502c46b1ce03ff14_l3.png)
![Rendered by QuickLaTeX.com \begin{array}{lll}\overrightarrow{ \mathstrut \text{OQ}}&=&\dfrac12\overrightarrow{ \mathstrut a}+\dfrac12(-s-t)\overrightarrow{ \mathstrut a}+s\overrightarrow{ \mathstrut b}+t\overrightarrow{ \mathstrut c}\\&=&\dfrac12(1-s-t)\overrightarrow{ \mathstrut a} +s\overrightarrow{ \mathstrut b}+t\overrightarrow{ \mathstrut c} \cdots\maru6\end{array}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-ef467cbd1689f7f7bfa37aa71f74b135_l3.png)
4点O, A, B, Cは同一平面上にないので,
![Rendered by QuickLaTeX.com \maru1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c8b8ec9c0d15342374d474f3407d687d_l3.png)
![Rendered by QuickLaTeX.com \maru6](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-4e34103dfe8448fbd33236d783bdf98f_l3.png)
![Rendered by QuickLaTeX.com \begin{cases}\dfrac15k=\dfrac{1-s-t}{2}\\\dfrac25k=s\\\dfrac25k=t\end{cases}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-2361bcbf1f48b5f7076bc5624205cde4_l3.png)
これを解いて,
![Rendered by QuickLaTeX.com k=\dfrac56](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-100e1715d5d46c42591d69abef5e11bc_l3.png)
![Rendered by QuickLaTeX.com s=\dfrac13](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-75e4671624e93eda51e0dae216e33e3c_l3.png)
![Rendered by QuickLaTeX.com t=\dfrac13](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5266db7643e5fc07c225d25da1d0d769_l3.png)
よって,
![Rendered by QuickLaTeX.com \overrightarrow{\text{OQ}}=\dfrac16\overrightarrow{ \mathstrut a}+\dfrac13\overrightarrow{ \mathstrut b}+\dfrac13\overrightarrow{ \mathstrut c}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-70a8fc984600baa9dbb23782e29527e9_l3.png)
(3) (2)より,
![Rendered by QuickLaTeX.com k=\dfrac56](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-100e1715d5d46c42591d69abef5e11bc_l3.png)
![Rendered by QuickLaTeX.com \bekutoru{OQ}=\dfrac56\bekutoru{OP}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-bd49fd8c297aaad7bc3bbc8a5ade985b_l3.png)
これより, OQ : OP
![Rendered by QuickLaTeX.com = 5 : 6](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-360eae816745ae11064cc90bd7ed7c3a_l3.png)
したがって, OQ : QP
![Rendered by QuickLaTeX.com = 5 : 1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-e6a57bb349c7334ab5f624db86b61643_l3.png)
ここがポイント
所定のベクトルを2通りの表し方で表して, 連立方程式を解いて求める。